Solve sqrt{7}-5sqrt{7}+3(sqrt{28}-sqrt{7/3})

$$\sqrt{ 7 } -5 \sqrt{ 7 } +3( \sqrt{ 28 } - \sqrt{ \frac{ 7 }{ 3 } } )$$

Evaluate

$$2\sqrt{7}-\sqrt{21}\approx 0.708926927$$

Short Solution Steps

Combine \sqrt{7} and -5\sqrt{7} to get -4\sqrt{7}.

$$-4\sqrt{7}+3\left(\sqrt{28}-\sqrt{\frac{7}{3}}\right)$$

Factor 28=2^{2}\times 7. Rewrite the square root of the product \sqrt{2^{2}\times 7} as the product of square roots \sqrt{2^{2}}\sqrt{7}. Take the square root of 2^{2}.

$$-4\sqrt{7}+3\left(2\sqrt{7}-\sqrt{\frac{7}{3}}\right)$$

Rewrite the square root of the division \sqrt{\frac{7}{3}} as the division of square roots \frac{\sqrt{7}}{\sqrt{3}}.

$$-4\sqrt{7}+3\left(2\sqrt{7}-\frac{\sqrt{7}}{\sqrt{3}}\right)$$

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$-4\sqrt{7}+3\left(2\sqrt{7}-\frac{\sqrt{7}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)$$

The square of \sqrt{3} is 3.

$$-4\sqrt{7}+3\left(2\sqrt{7}-\frac{\sqrt{7}\sqrt{3}}{3}\right)$$

To multiply \sqrt{7} and \sqrt{3}, multiply the numbers under the square root.

$$-4\sqrt{7}+3\left(2\sqrt{7}-\frac{\sqrt{21}}{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{7} times \frac{3}{3}.

$$-4\sqrt{7}+3\left(\frac{3\times 2\sqrt{7}}{3}-\frac{\sqrt{21}}{3}\right)$$

Since \frac{3\times 2\sqrt{7}}{3} and \frac{\sqrt{21}}{3} have the same denominator, subtract them by subtracting their numerators.

$$-4\sqrt{7}+3\times \frac{3\times 2\sqrt{7}-\sqrt{21}}{3}$$

Do the multiplications in 3\times 2\sqrt{7}-\sqrt{21}.

$$-4\sqrt{7}+3\times \frac{6\sqrt{7}-\sqrt{21}}{3}$$

Cancel out 3 and 3.

$$-4\sqrt{7}+6\sqrt{7}-\sqrt{21}$$

Combine -4\sqrt{7} and 6\sqrt{7} to get 2\sqrt{7}.

$$2\sqrt{7}-\sqrt{21}$$